Hybrid spatial and circuit optimization for targeted performance of mri coils

ABSTRACT

A method of operating a multi-coil magnetic resonance imaging system is disclosed which includes a controller performing a simulation using a predefined tissue model, determining output values of a variable of interest (VOI) associated with operation of two or more coils of an MRI system based on the simulation, comparing the simulated output values of the VOI to an a priori target values of the VOI, if the simulated output values of the VOI are outside of a predetermined envelope about the a priori target values of the VOI, then performing an optimization, wherein the optimization includes iteratively adjusting the circuit values until the simulated output values of the VOI are within the predetermined envelope about the a priori target values of the VOI thereby establishing VOI optimized values, and loading the established VOI optimized values and operating the magnetic resonance imaging system on the tissue to be imaged.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation application of the U.S.Non-Provisional patent application having Ser. No. 17/667,480 titled“Hybrid Spatial and Circuit Optimization for Targeted Performance of MRICoils” which was filed Feb. 8, 2022, which is related to and claims thepriority benefit of U.S. Provisional Patent Application having Ser. No.63/146/713 titled “Hybrid Spatial and Circuit Optimization for TargetedPerformance of MRI Coils” which was filed Feb. 8, 2021, the contents ofeach of which are hereby incorporated by reference in its entirety intothe present disclosure.

STATEMENT REGARDING GOVERNMENT FUNDING

This invention was made with government support under contract numbersEB011639, EB024408, EB026231 and NS090417 awarded by the NationalInstitutes of Health. The government has certain rights in theinvention.

TECHNICAL FIELD

The present disclosure generally relates to magnetic resonance imaging(MRI) systems, and in particular, to an MRI system with a multi-channeltransmit coil and a method of optimization of performance of same.

BACKGROUND

This section introduces aspects that may help facilitate a betterunderstanding of the disclosure. Accordingly, these statements are to beread in this light and are not to be understood as admissions about whatis or is not prior art.

Magnetic Resonance Imaging (MRI) has been a hallmark of imagingbiological tissues (e.g., a human brain) for decades, and has beenutilized ubiquitously. Reference is made to U.S. Pat. Pub. 2015/0028869in which a list of references are cited (U.S. Pat. Nos. 7,573,270;7,501,823; 7,358.923; 7,358,923; 7,345,485; 7,298,145; 7,285,957;7,173,425; 7,088,104; 7,088,100; 7,012,429; 6,940,466; 6,853,193;6,771,070; 6,552,544; 6,538,442; 6,107,798; 6,011,395; 5,998,999;5,791,648; 5,642,048; 5,610,521; 5,565,779; 5,483,163; 5,483,158;5,473,252; 5,461,314; 5,365,173; 5,243,286; 5,196,797; 5,185,575;5,172,061; 5,159,929; 5,081,418; 4,926,125; 4,918,388; 4,885,539;4,879,516; 4,871,969; 4,820,985; 4,788,503; 4,783,641; 4,780,677;4,752,736; 4,751,464; 4,737,718; 4,731,584; 4,725,780; 4,721,915;4,129,822; 4,320,342; and 4,638,253), each of which is incorporated byreference in its entirety into the present disclosure. Further referenceis made to U.S. Pat. Pubs. 2012/0153951 and 20140152309, each of whichis also incorporated by reference in its entirety into the presentdisclosure.

As a general proposition, it is critically important to provide highlevel of coil performance. Coil performance is measured by the how muchRF magnetic field is delivered to the coil per unit power. This variableis also measured against RF field homogeneity, without which fast andreliable MR images are difficult to obtain. Nowadays, MRI systems usemulti-channel transmission coils (e.g. 16 channels). Coil excitation canoccur by a single power source that is then split into a severalchannels, or alternatively each channel is powered separately by anassociated unit. One challenge with such multi-channel transmissioncoils is EM coupling. It is well known to characterize thiselectromagnetic coupling by an S-parameter used to denote scattering.Generally, the S-parameter refers to how a network of circuit elementsmaking up a transmit coil system would respond to differing inputs. Forexample the S-parameter may represent response of the coil system to onefrequency input vs. another frequency input. It is important tounderstand the S-parameter is a complex number with associated magnitudeand phase.

In order to overcome the electromagnetic coupling challenge, there havebeen efforts to compensate for the coupling. No matter how theelectromagnetic coupling is compensated, it is important that to makethe compensation independent of coil loading.

Another challenge is the decreased magnetic field component (B₁ ⁺)homogeneity over one or more region of interest (ROI). In order toachieve better homogeneity, RF shimming is a common technique, whereamplitude and phase of excitation signals are adjusted for a given ROI,when using a multi-channel transmitter [e.g., see W Gilbert, K M., A T.Curtis, J S. Gati, L M. Klassen, and R S. Menon: “A radiofrequency coilto facilitate B₁ ⁺ shimming and parallel imaging acceleration in threedimensions at 7 T,” NMR Biomed, vol 24., pp 815-823, 2011.].

Recent regulatory clearance for clinical use of 7 Tesla MRI (7T MRI) hasled to increased interest in clinical ultra-high field (UHF)applications. However, to robustly achieve the expected increase insignal to noise ratio associated with UHF MRI systems, the RF challengesneed to be met, namely, problems with higher RF power, worse B₁ ⁺homogeneity, and increased tissue conductivity but decreasedpermittivity at higher frequency, all of which usually results inincreased specific absorption rate (SAR). The use of paralleltransmission (pTx) coils combined with techniques such as RF shimmingand parallel excitation can mitigate the effects of B₁ ⁺ spatialhomogeneity. In particular, RF shimming can provide improvement in B₁ ⁺efficiency while reducing peak local SAR. In the development of such pTxcoils, the need for accurate EM simulations for RF safety andperformance design is evident. Over the past several years, thestate-of-the-art for design and simulation of such coil arrays hasadvanced via circuit-domain co-simulation strategy to use theS-parameters from a single electromagnetic simulation with RF circuitanalysis for coil tuning and matching thereby saving significant time.More specifically, prior art work has described calculation of aclosed-form S-parameter matrix to accomplish these simulations. Thesemethods have been used to improve the prediction of local SAR in pTxcoils at 3 T, 7 T, and 10.5 T.

As mentioned above, a characteristic challenge to using pTx array isstrong interactions between coils, which are not mitigated bypreamplifiers as in receive arrays. The strong interactions betweencoils and the subject at high frequency also make achieving reliabledecoupling between elements difficult. In order to minimize thesecouplings, several approaches have been proposed including capacitivedecoupling, inductive decoupling, and other methods such as inducedcurrent elimination (ICE), resonant inductive decoupling (RID), anddipole-loop decoupling. These methods range from geometric overlap toadditional secondary resonant circuits that can minimize both real andimaginary terms in the impedance matrix. It is clear however thatindependent of the specific methodology of decoupling, given thedecoupling circuit's effect on power distribution, it is important toinclude its impact in the EM simulation and circuit analysis. However,with the simulation tools available, solutions that include thedecoupling circuits are rare. The dual-row head coil modeled by Adrianyand Hoffmann included transformer decoupling (TD) circuits modeled usingthe built-in toolbox offered in CST MICROWAVE STUDIO 2018. However, morecomplex features, such as the Q factor and isolated resonant frequencyof the TD circuits, are not well modeled and optimized in the EMsimulation or circuit analysis software.

Therefore, there remains an unmet need for methods that can assist inthe design and optimization of a decoupled transceiver array in MRIsystems.

SUMMARY

A method of operating a multi-coil magnetic resonance imaging system isdisclosed. The method includes establishing initial circuit values of adrive circuit, loading a tissue model associated with a tissue to beimaged, loading target values for a variable of interest (VOI)associated with operation of two or more coils of a magnetic resonanceimaging system, performing a simulation based on the established circuitvalues and the loaded tissue model, determining output values of the VOIbased on the simulation, comparing the simulated output values of theVOI to the loaded target values of the VOI, if the simulated outputvalues are outside of a predetermined envelope about the loaded targetvalues of the VOI, then performing a first optimization. The firstoptimization includes establishing a cost function based on the VOI, anditeratively minimizing the cost function by iteratively adjusting thecircuit values until the cost function changes between iterations isless than a predetermined threshold, re-simulating, and re-comparing thesimulated output values of the VOI to the loaded target values of theVOI until the simulated output values are within the predeterminedenvelope.

A drive system for a multi-coil magnetic resonance imaging system isalso disclosed. The system includes two or more coils utilized forimaging a tissue of interest, a drive circuit for driving the two ormore coils, a controller having a processor and software loaded ontangible memory adapted to perform: establish initial circuit values ofa drive circuit, load a tissue model associated with a tissue to beimaged, load target values for a variable of interest (VOI) associatedwith operation of two or more coils of a magnetic resonance imagingsystem, perform a simulation based on the established circuit values andthe loaded tissue model, determine output values of the VOI based on thesimulation, compare the simulated output values of the VOI to the loadedtarget values of the VOI, if the simulated output values are outside ofa predetermined envelope about the loaded target values of the VOI, thenperform a first optimization. The initial optimization includesestablish a cost function based on the VOI, and iteratively adjust thecost function by iteratively adjusting the circuit values until the costfunction changes between iterations is less than a predeterminedthreshold, re-simulating, and re-compare the simulated output values ofthe VOI to the loaded target values of the VOI until the simulatedoutput values are within the predetermined envelope.

BRIEF DESCRIPTION OF DRAWINGS

The patent or application file contains at least one drawing executed incolor. Copies of this patent or patent application publication withcolor drawing(s) will be provided by the Office upon request and paymentof the necessary fee.

FIG. 1A is a schematic of an magnetic resonance imaging (MRI) system forimaging brain tissue depicting a slotted RF shield.

FIG. 1B is a schematic of voltage feeds' orientations of top and bottomrow loop coil elements in an FDTD (e.g., XFdtd) setup for a

208×208 S-parameters calculation with arrows showing current direction,with each voltage feed set to 1 V, 50Ω with modulated Gaussian waveexcitations.

FIG. 1C provides schematics of top and bottom row loop coil elements ofFIG. 1B.

FIG. 1D provides schematics of Resonant inductive decoupling (RID) andtransformer decoupling (TD) circuits.

FIG. 2 is a schematic of a 208-port array representing a network system,showing how 8 forward waves from the RF amplifiers are split to 16forward waves by the splitter, to feed the 16 matching circuits that areconnected to the array elements.

FIGS. 3A and 3B are magnitude (dB) and phase plots of the S-parametermatrix at 298 MHz, and frequency sweep determined from the cost functionin Equation 11 (with B₁ ⁺ homogeneity optimization) for the Louis model(as provided in FIG. 3A) and the equivalent data for Louis butdetermined with Equation 15, provided below, without B₁ ⁺homogeneityoptimization (as provided in FIG. 3B).

FIG. 4A provides the reconstructed B₁ ⁺ magnitude profiles(reconstructed using Eq. 14, provided below) of all 16 coils loaded withthe Louis model, and wherein each coil is fed with 65.5 V peak forwardvoltage.

FIG. 4B is a graph of B₁ ⁺SD/Mean vs. inter-row Δ phase in degrees forinter-row coil element phase shifts from 0° to 80°.

FIGS. 5A and 5B depict B₁ ⁺ magnitude profiles of eight channels on oneaxial slice of the Louis model (first and second rows) and in vivo head(third and fourth rows) as provided in FIG. 5A; the B₁ ⁺ phase profilesare presented in the same order as in FIG. 5A as shown in FIG. 5B.

FIGS. 5C and 5D show absolute magnitude (FIG. 5C) and phase (FIG. 5D) ofthe profiles in FIGS. 5A and 5B.

FIG. 6A shows Sagittal B₁ ⁺ profiles of the RF-shimmed homogeneousdistributions for the Ella (top left), Louis (top right), Hanako (bottomleft), and Duke (bottom right) models.

FIG. 6B provides the corresponding coronal B₁ ⁺profiles for the fourhead models shown in FIG. 6A.

FIG. 7 provides RF-shimmed B₁ ⁺ profiles from 10 volunteers. Each columnincludes seven evenly spaced axial slices from 1 volunteer.

FIG. 8 shows a side-by-side comparison between simulation and in vivo B₁⁺ maps of Hanako and a volunteer with similar head size.

FIG. 9A is a block diagram describing steps performed by a controller(not shown) to carry out the method including simulation andoptimization; and operation of the system of the present disclosure.

FIG. 9B is a block diagram that provides a more in-depth view of thesimulation and optimization steps outlined in FIG. 9A.

FIG. 10 is an example of a computer system constituting the controller(not shown) of FIG. 9A that can carry out the steps of the system of thepresent disclosure.

DETAILED DESCRIPTION

For the purposes of promoting an understanding of the principles of thepresent disclosure, reference will now be made to the embodimentsillustrated in the drawings, and specific language will be used todescribe the same. It will nevertheless be understood that no limitationof the scope of this disclosure is thereby intended.

In the present disclosure, the term “about” can allow for a degree ofvariability in a value or range, for example, within 10%, within 5%, orwithin 1% of a stated value or of a stated limit of a range.

In the present disclosure, the term “substantially” can allow for adegree of variability in a value or range, for example, within 90%,within 95%, or within 99% of a stated value or of a stated limit of arange.

A novel method is disclosed herein that can assist in the design andoptimization of a decoupled transceiver array in magnetic resonanceimaging (MRI) systems. Towards this end, the present disclosureintroduces a closed-form S-parameter matrix of a transceiver thataccounts for the matching circuits, decoupling circuits, and lumpedcapacitors. Additionally, a hybrid circuit-spatial domain analysis isintroduced that uses a target cost function which includes both theS-parameters and B₁ ⁺ homogeneity to determine coil parameters,including capacitors, inductors, and decoupling circuits' Q factors,isolated frequencies, and coupling coefficients. Over a series of foursimulated head models and an input range of coil parameters determinedfrom experimental data, this hybrid circuit-spatial domain analysisobtains excellent inter-subject consistency and agreement of actualcomponents. Finally, using the applied amplifier voltages from the MRIconsole, we generate B₁ ⁺ profiles from individual coils which show goodagreement with the in vivo data.

To achieve these novel features, the present disclosure presents aclosed-form equation of the coil S-parameters and overall spatial B₁ ⁺field, then introduce a cost function associated with the coilS-parameters and the B₁ ⁺ homogeneity in a subject's tissue (e.g., thebrain tissue), and then minimizing the cost function by optimizingtransceiver components, including matching, decoupling circuits, andlumped capacitors. Thereafter, the present disclosure provides acomparison in silico results determined with and without B₁ ⁺homogeneity weighting. Using the known voltage range from the hostconsole, the present disclosure thereafter reconstructs the B₁ ⁺ maps ofthe array coil and provides an RF shimming with four realistic headmodels. As performed with B₁ ⁺ homogeneity weighting, the optimized coilcircuit components were highly consistent over the four heads, producingwell-tuned, matched, and decoupled coils. The mean peak forward powersand B₁ ⁺ statistics in the head models are consistent with in vivo humanresults (n=8). There are systematic differences in the transceivercomponents as optimized with or without B₁ ⁺ homogeneity weighting,resulting in an improvement of 28.4±7.5% in B₁ ⁺ homogeneity with asmall 1.9±1.5% decline in power efficiency. Consequently, theco-simulation methodology presented herein accurately simulates thetransceiver, predicting consistent S-parameters, component values and B₁⁺ field. RF shimming of the calculated field maps match with in vivoperformance

An example MRI system with a double row array coil was modeled in XFdtd(v7.7, REMCOM, STATE COLLEGE, PA)—however, it should be understood thatother FDTD software can also be used, in 1-mm nominal cell resolution asshown in FIG. 1A, which is a schematic of an MRI system for imagingbrain tissue. FIG. 1A, depicts a slotted RF shield that is placedoutside the array. The dual-row coil array is made with a former andcovered with copper clad board. The simulated human models arepositioned in the array center, with the eyes aligned with the eyeportals on the array former, mimicking the real scanning scenario. Acylindrical RF shield was used which is made by two layers of overlappedslotted copper foil that are insulated from each other using a thinliquid crystal polymer layer with a relative permittivity of ε_(r)=2.The slotted copper foil produces a high-pass structure and provideseffective RF shielding. The coil former is modeled as polycarbonate(ε_(r)=3), and the board material of the coil is Bakelite (ε_(r)=3.5).The volume enclosed by the RF shield is meshed with 1-mm resolutiongridding; the rest of the space (including the human body and simulationpaddings) are meshed with gridding resolution of at least 20 cells perwavelength. A Japanese head model HANAKO (2-mm resolution adult female),two Virtual Family models ELLA (v1.3 1-mm resolution 26 year old female)and DUKE (v1.3 1-mm resolution 34 year old male), and the VirtualPopulation model LOUIS (v1.3 1-mm resolution 14 year old male) wereloaded separately in the coil center, with coil-to-tissue distancegreater than 1 cm. A broadband excitation with convergence criterion of−50 dB resulted a simulation time of 14 minutes per voltage port of amulti-port simulation, running on an Intel Xeon workstation with 64 GBRAM and two QUADRO K5200 GPUs.

The array coil (see FIG. 1A) has 208 gaps distributed along the loopsand is described by 296 parameters (a listing is shown in Table 1, firstcolumn). The voltage feeds are bridged across the gaps in theorientation as shown in FIG. 1B, which is a schematic of voltage feeds'orientations in the XFdtd setup for a

208×208 S-parameters calculation with arrows showing current direction,with each voltage feed set to 1 V, 50Ω with modulated Gaussian waveexcitations. In the circuit-domain, this array can be modeled as anetwork of 208 ports, with 16 ports connected to the matching circuits,and 192 ports connected to 112 lumped capacitors and 40 decouplingcircuits (using 2 ports each). At a given frequency, the

S-matrix is

$\begin{matrix}{{S_{CoilPorts} = \begin{bmatrix}S_{{drive}{drive}} & S_{{drive}{lump}} \\S_{{lump}{drive}} & S_{{lump}{lump}}\end{bmatrix}},} & \lbrack 1\rbrack\end{matrix}$

where the basic overall system equation is provided below in [4], below,describing the forward wave and reflected wave with the forward wavebeing defined in [2], below, and the reflected wave being defined in[3], below, wherein a_(drive) and b_(drive) are column vectors thatcontain 16 complex elements, annotated with complex vector space

. Specifically, a_(drive) and b_(drive) represent the forward andreflected waves connected to the driving ports on the coils,respectively, a_(lump) and b_(lump) are of

, representing the forward and reflected waves to the lumped capacitorsand decoupling circuits ports.

Referring to FIG. 1C, schematics of the top and bottom row loop coilelements are provided. Referring to FIG. 1D, schematics of Resonantinductive decoupling (RID) and transformer decoupling (TD) circuits areprovided. In total, 16 RID circuits are applied to decouple neighboringcoil elements in the horizontal direction, 8 vertical TD circuits areapplied to decouple neighboring elements in the vertical direction, and16 diagonal TD circuits are applied to decouple neighboring elements inthe diagonal direction.

With the experimental values for forward and reflected voltages beingmeasured at the amplifier, in this analysis the S-parameters aredescribed at the RF amplifiers, as shown in FIG. 2 , defined asb_(out)/a_(in) where a_(in) and b_(out) are the complex forward andreflected voltages, respectively, between the matching circuits andamplifiers. FIG. 2 is a schematic of a 208-port array representing anetwork system, showing how 8 forward waves from the RF amplifiers aresplit to 16 forward waves by the splitter, to feed the 16 matchingcircuits that are connected to the array elements. The array elementsare also connected with lumped capacitors and transformer decouplingcircuits. Here we show the analysis relating the coil elements used inthe optimization to the S-parameters b_(out)/a_(in).

The forward wave to the network system is a column vector expressed by:

a=[a ₁ a ₂ . . . a ₂₀₈]^(T) =[a _(drive) ^(T) a _(lump) ^(T)],  [2]

and the reflected waves of the network system are represented by acolumn vector:

b=[b ₁ b ₂ . . . b ₂₀₈]^(T) =[b _(drive) ^(T) b _(lump) ^(T)],  [3]

The overall relationship of all ports is:

$\begin{matrix}{{\begin{bmatrix}S_{{drive}{drive}} & S_{{drive}{lump}} \\S_{{lump}{drive}} & S_{{lump}{lump}}\end{bmatrix}\begin{bmatrix}a_{drive} \\a_{lump}\end{bmatrix}} = {\begin{bmatrix}b_{drive} \\b_{lump}\end{bmatrix}.}} & \lbrack 4\rbrack\end{matrix}$

TABLE 1 Mean ± SD of key components of the 296 optimized parareters(parameter numberings are not in ascending order) Hanako (excludeComponents Hanako B

 homogeneity) Ella Duke Louis

 Fixed lumped caps (10 pF or 8.2 pF) optimization 9.66 ± 1.24 9.52 ±0.87 9.66 ± 0.91 9.53 ± 0.85 9.69 ± 0.87 subject to

7, 13 pF

 Tuning cap,

 coils [10, 20 pF] 14.19 ± 0.85  13.75 ± 0.50  14.80 ± 0.24  14.19 ±0.40  14.37 ± 0.56 

 Tuning cap, bottom coils [10, 20 pF] 15.70 ± 1.53  15.11 ± 0.45  15.43± 0.39  15.52 ± 0.71  15.03 ± 0.55 

 Trimmer cap, matching, top coils [5, 20 pF] 5.56 ± 0.30 7.28 ± 0.706.58 ± 1.01 6.61 ± 1.05 6.02 ± 0.46

 Trimmer cap, matching, bottom coils [5, 20 pF] 10.84 ± 2.33  7.82 ±0.51 8.39 ± 2.21 6.97 ± 1.08 7.72 ± 1.29

 S

 matching cap [5, 20 pF] 6.22 ± 0.82 6.53 ± 0.56 6.38 ± 0.79 5.92 ± 0.436.18 ± 0.47

 Parallel matching cap [5, 25 pF] 18.93 ± 4.08  19.55 ± 0.58  19.93 ±2.24  20.16 ± 2.69  20.14 ± 2.34 

 RID inductor, top coils [5, 15 nH] 9.91 ± 1.05 9.34 ± 1.42 9.37 ± 0.9010.00 ± 1.46  9.55 ± 1.42

 RID inductor, bottom coils [5, 15 nH] 11.50 ± 1.20  9.49 ± 1.49 10.31 ±1.06  11.35 ± 1.30  10.18 ± 1.63 

 RID isolated frequency [200, 298 MHz] 290.27 ± 2.02  292.64 ± 0.87 291.48 ± 1.02  290.44 ± 1.64  291.46 ± 1.72 

 RID Q factors [150, 350] 235.46 ± 22.06  216.98 ± 15.90  238.24 ± 9.23 245.25 ± 16.26  236.94 ± 17.22 

 RID k coefficients [0.06, 0.5] 0.282 ± 0.029 0.257 ± 0.019 0.279 ±0.014 0.295 ± 0.021 0.280 ± 0.024

 TD vertical inductors [5, 20 nH] 17.93 ± 0.49  18.47 ± 0.33  17.81 ±0.26  17.93 ± 0.29  17.79 ± 0.32 

 TD diagonal inductors [5, 20 nH] 8.29 ± 1.38 9.98 ± 0.78 8.77 ± 1.539.77 ± 2.99 8.97 ± 1.72

 TD Q factors [150, 350] 249.95 ± 3.32  247.76 ± 1.59  249.02 ± 1.88 250.02 ± 1.42  248.92 ± 1.97 

 TD vertical k coefficients [0.06. 0.5] 0.424 ± 0.017 0.441 ± 0.0140.435 ± 0.012 0.416 ± 0.016 0.433 ± 0.014

 TD diagonal k coefficients [0.06, 0.5] 0.243 ± 0.018 0.255 ± 0.0130.256 ± 0.011 0.259 ± 0.023 0.255 ± 0.016 Note: The square bracketsindicate the minimum and maximum values for each parameter. The fixed

 capacitors are either

-pF or 8.2-pF capacitors on the actual co

.

indicates data missing or illegible when filed

The reflected waves can be represented by a function of reflectioncoefficients and forward waves as described by Lemdiasov (see LemdiasovRA, Obi AA, Ludwig R. A numerical postprocessing procedure for analyzingradio frequency Mill coils. Concepts Magn Reson Part A 2011). Eq. 5relates the reflected waves in relation to both the coil array and thematching circuits, as provided below:

$\begin{matrix}{\begin{bmatrix}b_{drive} \\b_{lump}\end{bmatrix} = {\begin{bmatrix}{S_{{match} +}^{- 1} \cdot a_{drive}} \\{S_{lump}^{- 1} \cdot a_{lump}}\end{bmatrix} - {\begin{bmatrix}{S_{{match} + -} \cdot S_{{match} +}^{- 1} \cdot a_{in}} \\0\end{bmatrix}.}}} & \lbrack 5\rbrack\end{matrix}$

The S_(match+) and S_(match+−) are diagonal matrices for the 16 matchingcircuits: the diagonal terms are the reflection and the transmissioncoefficients respectively, when looking from the loop coil toward thematching circuit. In Eq. 5, S_(lump) contains the S matrices of lumpedcapacitors (S_(cap)) and decoupling circuits (S_(DC)) described below:

$\begin{matrix}{S_{lump} = \begin{bmatrix}S_{cap} & 0 \\0 & S_{DC}\end{bmatrix}} & \lbrack 6\rbrack\end{matrix}$

Next, the S matrices for matching circuits (S_(match+), S_(match+−) andS_(match−)), lumped capacitors (S_(cap)) and decoupling circuits(S_(DC)) are described. The Z matrix for a given matching circuit iswritten as (using the capacitor notation in FIG. 2 ):

$\begin{matrix}{Z_{match} = {\begin{bmatrix}S_{{match} +} & S_{{match} + -} \\S_{{match} + -} & S_{{match} -}\end{bmatrix} = {{\begin{bmatrix}{\frac{- j}{\omega C_{s}} + \frac{\frac{j}{C_{s}C_{s}}}{\frac{2\omega}{C_{p}} + \frac{\omega}{C_{s}} + \frac{\omega}{C_{m}}}} & {- \frac{\frac{j}{C_{m}C_{s}}}{\frac{2\omega}{C_{p}} + \frac{\omega}{C_{s}} + \frac{\omega}{C_{m}}}} \\{- \frac{\frac{j}{C_{m}C_{s}}}{\frac{2\omega}{C_{p}} + \frac{\omega}{C_{s}} + \frac{\omega}{C_{m}}}} & {\frac{- j}{\omega C_{m}} + \frac{\frac{j}{C_{m}C_{m}}}{\frac{2\omega}{C_{p}} + \frac{\omega}{C_{s}} + \frac{\omega}{C_{m}}}}\end{bmatrix}.}}}} & \left\lbrack {6\_ 1} \right\rbrack\end{matrix}$

The S_(match+) and S_(match+−) are diagonal matrices for the 16 matchingcircuits: the reflection coefficient between the coil and matchingcircuit, and the transmit coefficient between the coil and voltage feedsrespectively. S_(match−) is the reflection coefficient between thevoltage feed (RF amplifier) and matching circuit. Thus, for the 16matching circuits:

$\begin{matrix}{S_{{match} +} = \begin{bmatrix}S_{{1{match}} +} & \ddots & 0 \\ \ddots & S_{{n{match}} +} & \ddots \\0 & \ddots & S_{{16{match}} +}\end{bmatrix}} & \left\lbrack {6\_ 2} \right\rbrack\end{matrix}$ And $\begin{matrix}{S_{{match} + -} = {\begin{bmatrix}S_{{1{match}} + -} & \ddots & 0 \\ \ddots & S_{{n{match}} + -} & \ddots \\0 & \ddots & S_{{16{match}} + -}\end{bmatrix}.}} & \left\lbrack {6\_ 3} \right\rbrack\end{matrix}$

Returning to equation 5, inserting Eq. 5 in Eq. 4 provides:

$\begin{matrix}{{\begin{bmatrix}S_{{drive}{drive}} & S_{{drive}{lump}} \\S_{{lump}{drive}} & S_{{lump}{lump}}\end{bmatrix}\begin{bmatrix}a_{drive} \\a_{lump}\end{bmatrix}} = {{\begin{bmatrix}{S_{{match} +}^{- 1} \cdot a_{drive}} \\{S_{lump}^{- 1} \cdot a_{lump}}\end{bmatrix} - {\begin{bmatrix}{S_{{mat{ch}} + -} \cdot S_{{mat{ch}} +}^{- 1} \cdot a_{in}} \\0\end{bmatrix}.}}}} & \lbrack 7\rbrack\end{matrix}$

To determine the array S-parameters from b_(out)/a_(in), with b_(out)given by the relationship between the matching circuits' S_(match)parameters and a_(drive), b_(out) is expressed as:

$\begin{matrix}{{b_{out} = {{S_{{match} + -} \cdot \frac{a_{drive} - {S_{{match} + -} \cdot a_{in}}}{S_{{match} +}}} + {S_{{match} -} \cdot a_{in}}}},} & \lbrack 8\rbrack\end{matrix}$

where a_(drive) is calculated from Eq. 7 as

a _(drive)=(S _(match+) ⁻¹ −S _(drive drive) −S _(drive lump)·(S _(lump)⁻¹ −S _(lump lump))⁻¹ ·S _(lump drive))⁻¹ ·S _(match +−) ·S _(match+) ⁻¹·a _(in).  [9]

Thus the

S-parameters of the array coil measured at the RF amplifier is expressedby:

$\begin{matrix}{S = {\frac{b_{out}}{a_{in}} = {{S_{{match} + -} \cdot S_{{ma{tch}} +}^{- 1} \cdot \left( {\frac{a_{drive}}{a_{in}} - S_{{match} + -}} \right)} + {S_{{match} -}.}}}} & \lbrack 10\rbrack\end{matrix}$

To validate Eq. 10, the XFdtd-calculated S_(coilPorts) from Eq. 1 wasconverted to TOUCHSTONE file and imported into the N-port S-parameterinstance in ADVANCED DESIGN SYSTEM (ADS 2020, KEYSIGHT, Santa Rosa, CA)and connected with corresponding circuit models (lumped capacitors,matching circuits, and decoupling circuits). The resulting S-parametersobtained at the 16 feed ports in the ADS 2020 N-port S-parameterinstance are the same as the

S-parameters obtained using Eq. 10.

In the above 208-port array description, a lumped component descriptionof the decoupling circuits was utilized. FIG. 1B shows the adjacentsurface coils being decoupled using two types of decoupling circuits.Depending on the orientation, the nearest neighbor surface coils havemutual impedance Z₁₂=−jωM₁₂+R₁₂, where ω is coil resonant angularfrequency 2π×298 MHz. As shown in FIG. 1B, within a row (top or bottom),the horizontally adjacent coils are decoupled by RID circuits; thevertically and diagonally (“cornering”) adjacent coils are decoupled byTD circuits (see FIG. 1D). For the RID circuit, both the reactive andresistive terms can be canceled when ω₀<ω while the TD eliminates thereactive term of the mutual impedance.

Referring back to Eq. 6, S_(lump) is of

and contains the S-parameters of lumped capacitors (S_(cap)) anddecoupling circuits (S_(DC)), where S_(DC) is of

and contains the 40 decoupling circuits' S-parameters:

$\begin{matrix}{{S_{DC} = \begin{bmatrix}S_{{DC}1} & \ddots & 0 \\ \ddots & S_{{DC}n} & \ddots \\0 & \ddots & S_{{DC}{}40}\end{bmatrix}},} & \left\lbrack {10\_ 1} \right\rbrack\end{matrix}$

where S_(DC n) is a

S-parameters of the n^(th) decoupling circuit. Since each decouplingcircuit has 2 ports, we have:

$\begin{matrix}{{S_{{DC}n} = {\begin{bmatrix}S_{11} & S_{12} \\S_{21} & S_{22}\end{bmatrix} = {\left( {Z_{{DC}n} - {50I}} \right)\left( {Z_{{DC}n} + {50I}} \right)^{- 1}}}},} & \left\lbrack {10\_ 2} \right\rbrack\end{matrix}$

where the Z_(DC n) is a

impedance matrix of the n^(th) RID or TD circuit, and their expressionsare calculated as below. As shown in FIG. 1B, the RID decouplingcircuits are used to decouple within-row adjacent coil pairs, while theTD circuits are used to decouple “cornering” coils and between-row coilpairs. Using notations in FIG. 1D, The Z matrix (

) of each RID circuit expressed as:

$\begin{matrix} & \left\lbrack {10\_ 3} \right\rbrack\end{matrix}$ $Z_{RID} = {{\begin{bmatrix}\begin{matrix}{{j\omega L_{0}} + R_{0} -} \\\frac{\omega^{2}k_{0}^{2}{L_{0}^{2}\left( {{j\omega L_{0}} - \frac{j}{\omega C_{0}} + R_{0}} \right)}}{\left( {{j\omega L_{0}} - \frac{j}{\omega C_{0}} + R_{0}} \right)^{2} - \left( \frac{j}{\omega C_{0}} \right)^{2}}\end{matrix} & \frac{{- \omega^{2}}k_{0}^{2}{L_{0}^{2}\left( \frac{j}{\omega C_{0}} \right)}}{\left( {{j\omega L_{0}} - \frac{j}{\omega C_{0}} + R_{0}} \right)^{2} - \left( \frac{j}{\omega C_{0}} \right)^{2}} \\\frac{{- \omega^{2}}k_{0}^{2}{L_{0}^{2}\left( \frac{j}{\omega C_{0}} \right)}}{\left( {{j\omega L_{0}} - \frac{j}{\omega C_{0}} + R_{0}} \right)^{2} - \left( \frac{j}{\omega C_{0}} \right)^{2}} & \begin{matrix}{{j\omega L_{0}} + R_{0} -} \\\frac{\omega^{2}k_{0}^{2}{L_{0}^{2}\left( {{j\omega L_{0}} - \frac{j}{\omega C_{0}} + R_{0}} \right)}}{\left( {{j\omega L} - \frac{j}{\omega C_{0}} + R_{0}} \right)^{2} - \left( \frac{j}{\omega C_{0}} \right)^{2}}\end{matrix}\end{bmatrix},}}$

where

${C_{0} = \frac{2}{\omega_{0}^{2}L_{0}}},$

and

R ₀=ω₀ L ₀ /Q ₀.

The TD circuit Z matrix is written as:

$\begin{matrix}{{Z_{TD} = \begin{bmatrix}{{j\omega L_{1}} + R_{1}} & {j\omega k_{1}L_{1}} \\{j\omega k_{1}L_{1}} & {{j\omega L_{1}} + R_{1}}\end{bmatrix}},} & \left\lbrack {10\_ 4} \right\rbrack\end{matrix}$

where

R ₁=ω₀ L ₁ /Q ₁.

Thus, the Z matrix (

) of each RID circuit is characterized by four variables ω₀, L₀, Q₀ andk₀, representing the isolated resonant frequency, inductors, RID Qfactor and coupling coefficients. Consequently, the capacitor value (C₀)and the inductor resistivity (R₀) can be determined based on C₀=2/(ω₀²L₀), and R₀=ω₀L₀/Q₀. The Z matrix (

) of each TD circuit (see equation 10_4) is characterized by threevariables L₁, Q₁ and k₁, representing inductors, inductor Q factor andthe coupling coefficient. The inductor resistivity (R₁) of the TDcircuit is given by R₁=ωL₁/Q₁.

An optimization process is also disclosed herein. The optimization wasperformed using a cost function (ƒ(x)) of a real number vector x of

whose entries are coil parameters. The ƒ(x) is defined in Eq. 11, where∥ ∥ denotes the Euclidean distance, | | is the elementwise absolutevalues, and w₁₋₃ are weights:

$\begin{matrix}{{f(x)} = {{w_{1}{{{❘{{diag}\left( {S(x)} \right)}❘} - S_{ii}}}} + {w_{2}{{{❘{S_{r}(x)}❘} - S_{ij}}}} + {w_{3}{{{\frac{St{d\left( B_{1} \right)}}{{Mean}\left( B_{1} \right)} - {target}}}.}}}} & \lbrack 11\rbrack\end{matrix}$

The minimum is given by the constrained optimization

$\overset{\hat{}}{x} = {\underset{x}{argmin}\left\{ {f(x)} \right\}}$

over the 296 parameters, and each coil port has at least one parameter(see Table 1, left column for itemization for x), subject tox∈{Ω:x_(n lower)<x_(n)<x_(n upper), n=1,2, . . . , 296}.

Eq. 11 contains three parts, with the first part optimizing the diagonalterms of the S-parameters S(x) from Eq. 10 (denoted by diag(S(x))). TheS_(ii) is a column vector of

, and its elements are set to a target value chosen from −20 to −25 dB(the best value is −20 dB). The second part is optimization ofdecoupling of any two adjacent coils, represented by the selectedelements in the strictly lower triangle portion of the coil S-parameter(denoted by S_(r)(x), terms are shown as S_(ij(RID bot)),S_(ij(RID top)), and S_(ij(TD)) in FIGS. 3A and 3B frequency-sweepplots). FIGS. 3A and 3B are magnitude (dB) and phase plots of theS-parameter matrix at 298 MHz, and frequency sweep determined from thecost function in Equation 11 (with B₁ ⁺ homogeneity optimization) forthe Louis model (as provided in FIG. 3A) and the equivalent data forLouis but determined with Equation 15, without B₁ ⁺homogeneityoptimization (as provided in FIG. 3B). The S_(ij) is a column vector of

, and its elements are set to a target value chosen from −20 to −30 dB(the best value is −25 dB); these boundary values are based onexperimental data. Notably, S_(r)(x) are effectively defined by the TDand RID circuits (rather than the sample), so the optimization ofS_(r)(x) is equivalent to optimization of the TD and RID circuits. Thethird part is the optimization of B₁ ⁺ coefficient of variation (CV), ametric for B₁ ⁺ homogeneity within the intracerebral tissue calculatedusing Eqs. 9, 12-14 (see below); we used a target B₁ ⁺ CV of 10%. Theweightings w₁, w₂ and w₃ were set at 0.1, 0.1, and 1.5, based onobtaining equal contributions to the ƒ(x) minimization from the threetarget parameters. To avoid large matrix sizes and to save optimizationtime, we downsampled the B₁ ⁺ maps from 300² to 64² resolution over the9 slices (i.e., ˜20 fold reduction in matrix size) before calculatingStd(B₁ ⁺) and Mean(B₁ ⁺).

This constrained optimization can be solved using algorithms such asSelf Organizing Migrating Algorithm (SOMA), the alternating directionmethod of multipliers (ADMM) (the implementation is described inAppendix B), Genetic Algorithm (GA) and a nonlinear programming solver‘fmincon’ using the ‘interior-point’ algorithm. Both GA and fmincon areprovided in the MATLAB Optimization Toolbox (MathWorks, Natick, MA). Theoptimization performance of optimizing coil parameters in Eq. 11 arecompared between the four algorithms (SOMA, GA, ADMM and ‘fmincon’). Weused fmincon as the optimization solver and searched for optimal xparameters within the upper and lower bounds.

As shown in Table 1, over the four modeled heads Ella, Duke, Hanako andLouis, the constrained optimization of ƒ(x) gave highly consistentvalues for all mean component values (264 parameters associated with theRID, TD, and distributed capacitors, excluding tuning and matchingcircuits). These values were thus used to define the “fixed” transceiverT₀. In the experiment, each coil in the array is tuned and matched oneach subject. To mimic real-world workflow, T₀ was then used withoptimization of the 32 tuning and matching capacitors (x₉₇ to x₁₂₈) foreach head to estimate field maps and RF shim.

As shown in FIG. 2 , the forward waves from the 8-channel RF amplifierare split to 16 forward waves (a_(in)) to feed the 16 coil elements.While the RF amplifier of the SIEMENS MAGNETOM 7T VB17 8 pTx system iscapable of up to 165 V per channel, to target a final B₁ ⁺ of 11.74 μT,each channel is optimized with a maximum of 110 V. To include thesplitter and approximate real-world conditions, we applied 4.5 dBattenuation on the forward voltage, i.e., each coil received maximum65.5 V forward voltage. Thus, the expression of forward voltage (a_(in))for each coil at the coil plug-in is:

a _(in) =V _(k) e ^(j(ϕk)) =V ₀ e ^(j(ϕCP+) ^(Δ) ^(inter-row)),  [12]

where V_(k) is the forward voltage amplitude, andϕ_(k) is applied forward voltage phase (k=1, 2, . . . , 16). To generatea circularly polarized (CP) distribution, ϕ_(CP) is π/4 phase incrementsalong the 8 channels in the azimuthal direction. It should beappreciated that there is an additional phase shift Δ_(inter-row)(identical for all 8 channels) between the top and bottom rows of thearray, reflecting additional cable lengths associated with the splitter.An optimization of Eq. 11 over a range of Δ_(inter-row) (0-80°) showedthe best B₁ ⁺ homogeneity at >50° (see FIG. 4B which is graph of B₁⁺SD/Mean vs. inter-row Δ phase in degrees for inter-row coil elementphase shifts from 0° to 80°). We used Δ_(inter-row)=56° for theremainder of the work as presented herein.

The a_(in) is a vector concatenating a_(in) of all 16 coils.Substituting this a_(in) into Eq. 9 gives a_(drive), and using Eq. 13gives a_(lump).

a _(lump)=(S _(lump) ⁻¹ −S _(lump lump))⁻¹ ·S _(lump drive) ·a_(drive)  [13]

After calculating a_(drive) and a_(lump), we can obtain the forward wavevector a in Eq. 2, and its elements a_(n) are used in the following Eq.14. Here we can use Eq. 14 to generate B₁ ⁺ maps corresponding to thefollowing driving conditions: (A) simultaneous transmission through allcoils in CP-mode, using a_(in) that has all elements equal to thevoltages from Eq. 12; (B) single-coil transmission (16 channels), whereeach coil map is generated using a_(in) that has only one element equalto the voltage value, and the rest being zeros; (C) pairwisetransmission (8 channels), where each channel corresponds to a verticalpair of coils, generated using a_(in) that contains two elements equalto the voltage value, and the rest being zeros. The 8-channel B₁ ⁺mapsare later used to generate the optimized B₁ ⁺ distribution.

$\begin{matrix}{B_{1}^{+} = {\sum\limits_{n = 1}^{208}{a_{n} \cdot \frac{B_{1_{{voltage}{source}n}}^{+}}{a_{{voltage}{source}n}}}}} & \lbrack 14\rbrack\end{matrix}$

where a_(n) is the nth element of the forward wave vector a, the B₁_(voltage source n) ⁺ voltage source n is the B₁ ⁺ field map generatedby the n^(th) voltage feed (one of the 208 voltage feeds in XFdtd), anda_(voltage source n) is the forward voltage at the load of the n^(th)voltage feed. The a_(voltage source n) is calculated based on the loadvoltage and the reflection coefficient seen at the n^(th) voltage. Tocheck the accuracy of Eq. 14, a simplified coil array (without thedecoupling and matching circuits) was used, i.e., a direct simulationwas performed with XFdtd (although other FDTD software can be used)using 18.5 pF lumped capacitors and 1-volt voltage sources bridging thecoil gaps. The resulting S-parameters and field maps are compared to thecalculated co-simulated results obtained without decoupling and matchingcircuitry using Eqs. 10 and 14, respectively.

For the “Homogeneous” distribution that targets the intracerebralregion, two Regions of interest (ROIs) are defined: an inner ROI(ROI^(HomogPhase)) over which the phase per channel is calculated and alarger outer ROI (ROI^(HomogAmp)) that includes all of the intracerebraltissue. The mean phase in ROI^(HomogPhase) is subtracted from eachchannel's phase map to obtain a constant phase across all channels. Theamplitudes of the forward voltages of the 8 channels are optimized toachieve the targeted B₁ ⁺ (11.74 μT) in the ROI^(HomogAmp).

The experimental procedure is next described. For each subject, eachcoil in the array was tuned and matched using an MM compatible RFsweeper probe (MORRIS INSTRUMENTS INC., Ottawa CA). Due to the highdegree of decoupling of the coils within the array, tuning and matchingadjustments for individual coils were wholly independent, making theprocess non-iterative, achieved in 2-4 minutes (5-10 seconds per coil),with sufficient dynamic range to account for phantom and human headloading conditions.

The B₁ ⁺ maps were acquired on SIEMENS MAGNETOM 7T VB17 8 pTx systemusing the vendor provided acquisition routines, which generate relativeamplitude, phase and flip angle maps through: 1) a multi-slice gradientecho acquisition using a single transmit channel for each excitation and2) a FLASH sequence with an initial preparatory weighting pulsedelivered from a single channel. The B₁ ⁺ maps were acquired with a FOV240×240 mm², 64×64 resolution over 11 slices 5 mm thick/gap 5 mm. The B₁⁺ data was acquired as part of the routine calibrations performed in anongoing IRB approved study. n=10 subjects (6F), mean age 21.8+/−4.9.

Using the above described experimental procedures, the following resultswere obtained. Despite the variation in head sizes, the optimizationyielded very consistent results as shown in Table 1, with an overallcoil parameters CV of about 8%. Thus the “fixed” transceiver T₀ wasdefined from the mean values from the four heads for each component.Using T₀, we then optimized the tuning and matching capacitors togenerate the coil S-parameter matrix at single frequency 298 MHz foreach head model. The S-parameter matrices are shown in FIG. 3A, whichhave element-wise similar magnitude and phase over the four models. Forall four models at 298 MHz, the S_(ii) and S_(ij) (for adjacentdecoupled coil pairs) are close to their target values of −20 dB and −25dB, respectively, while the S_(ij) of the next adjacent coil (which isnot explicitly decoupled) are all better than −14 dB. The top row 8coils (the 9^(th) to 16^(th) coils) are more difficult to decouplebecause the top row coils are angled inward towards the human head,increasing the mutual resistance between adjacent coils. However, forall adjacent coils decoupled by the RID circuits, the S-parameterfrequency sweep showed a dip, reaching S_(ij) values of −30 dB to −50dB.

FIG. 4A shows the reconstructed B₁ ⁺ magnitude profiles (reconstructedusing Eq. 14) of all 16 coils loaded with the Louis model, and each coilis fed with 65.5 V peak forward voltage. Each coil has similar spatialdistribution of B₁ ⁺ magnitude, as each coil is well decoupled from theadjacent coils, with the coupling between the next nearest neighborcoils reaching −14 dB isolation (e.g., S13,11 and S14,12 in FIGS. 3A and3B S matrix magnitude map). Allowing for variation in Δ_(inter-row),FIG. 4B shows that two simulating heads behaved similarly for resultingB₁ ⁺ homogeneity variation, with an approximate minimum beyond >50.

FIGS. 5A and 5B depict B₁ ⁺ magnitude profiles of eight channels on oneaxial slice of the Louis model (first and second rows) and in vivo head(third and fourth rows) as provided in FIG. 5A; the B₁ ⁺ phase profilesare presented in the same order as in FIG. 5A as shown in FIG. 5B. Thephase map of each channel is relative to the first channel. In both thesimulation and experiment, each coil element is fed with 65.5-V peakforward voltage. The absolute magnitude is shown in FIG. 5C and thephase is shown in FIG. 5D. FIGS. 5A-5D show difference between the Louismodel and in vivo heads for each of the eight channels' B₁ ⁺ profileshave good agreement in the B₁ ⁺ maps (phase, amplitude) of individualchannels between the Louis model and experimental data. In bothsimulation and experiment, each of the 16 coils is fed with 65.5 V peakforward voltage. Results from the RF shimmed homogeneous B₁ ⁺ maps usingcost function Eq. 11 are shown in FIGS. 6A and 6B and in Table 2,finding reasonable agreement with in vivo data obtained from eight humansubjects.

FIG. 6A shows Sagittal B₁ ⁺ profiles of the RF-shimmed homogeneousdistributions for the Ella (top left), Louis (top right), Hanako (bottomleft), and Duke (bottom right) models. The nine region of interestslices indicated with dashed red lines are evenly spaced across thehead. FIG. 6B provides the corresponding coronal B₁ ⁺profiles for thefour head models.

FIG. 7 shows the RF-shimmed B₁ ⁺ profiles from 10 volunteers. Eachcolumn includes seven evenly spaced axial slices from 1 volunteer.

FIG. 8 shows a side-by-side comparison between simulation and in vivo B₁⁺ maps of Hanako and a volunteer with similar head size. Seven evenlyspaced, RF-shimmed B₁ ⁺ axial profiles of the Hanako model (firstcolumn) and in vivo heads (second column) are shown in FIG. 8 ; thecorresponding tissue maps for the Hanako model and in vivo heads areshown in the third and fourth columns, respectively. There is excellentagreement in B₁ ⁺ efficiency (mean B₁ ⁺ in brain divided by forwardpower), achieving an in silico 11.16±0.35 μT/√{square root over (W)}, incomparison in vivo 11.39 μT/√{square root over (W)}, and good agreementin B₁ ⁺ homogeneity, at 13.6±0.4% in silico, in comparison to ourprevious reports at 11-13% and 10.5±1.5% (n=8 subjects) in vivo.

TABLE 2 Mean and SD

 of

 in the intra

 tissue, and the total peak forward power of the RF amplifier for thehomogeneous distribution Peak forward B

 efficiency B

 mean (μT) B

 SD (μT) B

 SD/mean % power (W) (μT/√{square root over (W)}) A. With

 homogeneity in the

 function Hanako (

) 11.45 1.625 14.2 1920.8 0.2612 Ella (3.20 L) 11.47 1.564 13.6 1922.80.2616 Duke (3.75 L) 11.13 1.508 13.6 1936.0 0.2530 Louis (3.

 L) 11.53 1.508 13.1 1781.3 0.2732

 simulated 11.39 ± 0.18 1.550 ± 0.06 13.6 ± 0.4 1890.2 ± 72.9 0.2620 ±0.0082 In vivo (

) 11.10 ± 0.10 1.100 ± 0.17 10.5 ± 1.5

 ±

04 0.2675 B. Excluding

 homogeneity in the

 function Hanako 11.16 2.2

19.8 1705.6 0.2701 Ella 11.17 1.895 17.0 1782.8 0.2645 Duke 11.11 1.91417.2 1824.4 0.2600 Louis 11.34 1.832 16.2 1716.4 0.2736

 simulated 11.19 ± 0.10 1.964 ± 0.17 17.6 ± 1.6 1757.3 ± 56.3 0.2668 ±0.0061 C. Optimizing the

 and matching capacitors on the fixed transceiver T

Hanako 11.44 1.722 1

.1 1846.1 0.266

Ella 11.42 1.534 13.4 1931.0 0.2

98 Duke 11.39 1.571 13.8 1915.7 0.2602 Louis 11.53 1.510 13.1 1748.20.2757 Mean sim

11.45 ± 0.06 1.583 ± 0.09 13.8 ± 0.9 1

60.3 ± 83.3 0.2656 ± 0.0075 Note: The

 volumes inside the RF shield

 in the

 including

 by weighing

 excluding homogeneity

 is performed

 over

 the full set of

 (A) and

 The fixed transceiver

 used

 determined from the homo

 weighted

indicates data missing or illegible when filed

Table 2 shows the performance when using the T₀ transceiver on all fourhead models, optimizing only the tuning and matching capacitors. Asexpected, the B₁ ⁺ efficiency and B₁ ⁺ homogeneity are in good agreementwith in vivo data.

The decoupled array design is advantageous due to better control of thecoil interactions that affect homogeneity and amplitude. In thisanalysis, we were able to consider the homogeneity as a design featurein the cost function (Eq. 11) and examine the consequent impact on thecoil components. For this comparison, we modified the cost function toeliminate the homogeneity condition, giving Eq. 15:

ƒ(x)=w ₁∥|diag(S(x))|−S _(ii) ∥+w ₂ ∥|S _(r)(x)|−S _(ij)∥  [15]

where the elements in S_(ii) and S_(ij) are set to −20 dB and −40 dB,respectively.

As shown in Table 2, comparing the effect of homogeneity weighting, themean B₁ ⁺ CV worsened by 28.4±7.5% to 17.5±1.6%, while the B₁ ⁺efficiency is slightly improved. Hanako exhibited the greatest change,an absolute 5% drop in CV, 14.2%>19.8%, i.e., a ˜39% change inhomogeneity. The consequences of omitting the homogeneity weighting forHanako are seen throughout the decoupling circuits, with increasedisolated RID resonance frequency and TD inductor values, decreased Qfactors, RID inductor values, and RID k coefficients. FIGS. 3A and 3Bcompare the S-parameters and frequency sweeps calculated with andwithout homogeneity weighting for the Louis model. It is notable thatwith omission of the homogeneity weighting, the majority of change inthe S-parameters amplitude and phase are in the top-top and top-bottomrow coil interactions, with little effect in the bottom-bottom rowinteractions. Comparing with and without homogeneity weighting, thetop-top and top-bottom S_(ij) values are remarkably worse (largervalues) with homogeneity weighting. Nonetheless, with homogeneityweighting, this range of values for the S_(ij(RID top)),S_(ij(RID bot))and S_(ij(TD)) is similar over the four models.

The present disclosure, thus, provides a co-simulation method, pairedwith S-parameters and B₁ ⁺ homogeneity optimization to simulate adouble-row, 16-coil head transmit array at 7 T. Our co-simulation modelconsidered the matching circuits, decoupling circuits, and lumpedcapacitors. With the RID and TD circuits, the array coil can be tunedand matched at various loadings in silico with all coil elementsachieving S_(ij) coupling better than about −14 dB, consistent withknown in vivo performance. The optimization parameters accuratelycharacterized the decoupling circuits, e.g., the Q factors and isolatedresonant frequencies of decoupling circuits are similar to thosepreviously published, and would be important to account for the effecton RF power distribution by the decoupling circuits. Based on thisco-simulation, the coil S-parameters and B₁ ⁺ homogeneity can beoptimized by different constrained optimization algorithms (ADMM, SOMA,GA, and fmincon). The resulting complex field maps of individual andsummed coils show excellent agreement with in vivo data.

In the present disclosure a comparison of the behavior of theS-parameters using two cost functions that explicitly use spatialinformation (Eqs. 11 and 15) is provided. Comparing results with andwithout the homogeneity cost function, there were a significant 23% and38% changes in the trimmer capacitor values on the matching circuit oftop and bottom row coils respectively, and 21% change in the inductorvalues of the RID circuit on the bottom row coils. It is of interestthat the homogeneity weighting worsens the S_(ij) values, particularlyaffecting the top row coils and their coupling with the bottom row. Thismay reflect a penalty on power efficiency (1.9±1.5% decline, Table 2) inorder to improve B₁ ⁺homogeneity. In this manner, the inclusion of thehomogeneity cost function is effectively making use of both themagnitude and phase of the S-parameters, the phase which is commonlyignored in RF simulation studies.

Over the four heads placed in the coil center, the simulation generatedhighly consistent values for the component terms (Table 1). The headvolumes inside the RF shield are reported in Table 2 left column.Several observations are of note. First, even though there is a 16.3%difference in head volume between the Duke head (3.75 liters, determinedfrom all head tissue within the RF shield) and Hanako head (3.14liters), there was no significant difference between any matchingcircuit or decoupling components and minimal differences in S_(ij),indicating that with the applied decoupling circuits, the residualimpedance is small. Second, the effects of the decoupling circuits areclear. As demonstrated by the validation simulation of Eq. 14, for asingle activated coil at bottom row, the highest S_(ij) about −8 dB) iswith adjacent coil, significantly worse than the scenarios after addingthe RIDs where the highest S_(ij) about −15 dB) is with the nextadjacent coils.

There is also good agreement between simulated and in vivo B₁ ⁺ profilesand RF power efficiency. In RF shimming, we achieved a mean B₁ ⁺ CV of13.6±0.4% and B₁ ⁺ efficiency 11.16±0.35 μT/√{square root over (W)} incomparison to experimental data here of B₁ ⁺ CV 10.5±1.5%, B₁ ⁺efficiency 11.37±0.26 μT/√{square root over (W)}, and of 11-13% CVpreviously reported, Table 2. These residuals are the result ofdifferences in head size and anatomical geometry which can affect thesize of the intracerebral tissue ROIs and RF field propagation; forexample, a less heterogeneous CSF distribution in brain result in lessheterogeneous tissue conductivity and consequently less eddy currentshielding. Residual differences can be a caused by head position or tiltwithin the array, or may still be affected by the accuracy of the modelheads (e.g., in mesh size or tissue properties; for example, suchmeasurements may depend on the temperature of the tissue, in vivo or invitro.

Consequently, the improved correspondence of simulated and in vivo B₁ ⁺maps can reduce the model error for the array. A reduced model errorwill reduce the safety factor which is used to account forunderestimation of the worst-case SAR. The present disclosure, thus,presents a hybrid circuit-spatial domain and cost function optimizationto accelerate the FDTD simulation and design stages of a double row pTxhead coil. The resulting field maps are in excellent agreement with invivo results, and the high consistency of the coil components (typicallyvarying by 2 to 8%, mean 3.4%) over the 4 simulated heads contends thatthe methods are robust and identify realistic component values. Theinclusion of the spatial B₁ ⁺ homogeneity into the cost function isnovel and demonstrates that the optimization of this decoupled array isbased on the desired homogeneity, amplitude, and power efficiency. Theavailable solution space shows that a substantial gain in homogeneity(28%) can be achieved with a near-negligible (2%) loss in amplitude andefficiency. From a coil design view, the methods presented herein arenot limited by the complexity of the coil designs such as the number oflumped components, nor coil-to-coil proximity, and should thus beapplicable to analysis and simulation of array coil designs of higherport counts and geometrically overlaid coils. Therefore, no limitationshould be applied to the specifics of the experimental proceduresprovided herein.

Referring to FIG. 9A, a block diagram is provided showing the multi-coilsystem and the method of operation of same of the present disclosure,according to one embodiment. Specifically, the block diagram shown inFIG. 9A describes steps performed by a controller (not shown) to carryout the method including simulation and optimization; and operation ofthe system of the present disclosure. A controller (not show) beginswith setting values for a drive circuit (initial values). Then thecontroller loads a tissue model as described above, followed by loadinga target value for a variable of interest (VOI). The VOI can be any oneof cross-talk, B₁ ⁺ homogeneity, B₁ ⁺ power efficiency, and anycombination thereof. The controller (not shown) then optimizes based ona cost function (see Eq. 11). Iteratively, the controller (not shown)inquires whether the VOI has reached values within an acceptable rangeof the target. If not, then the drive circuit values are adjusted andthe simulation is re-run. This iterative process constitutes the initialoptimization, according to the present disclosure. A more detailrepresentation of the simulation and optimization engine is presented inFIG. 9B, described below. Once the circuit parameters have beenoptimized for the VOI, the controller (not shown) exports circuitvalues, B, E, and S-parameters. These parameters are utilized tofabricate the drive circuit, and to setup the coils. Next, thecontroller (not shown) is used to run an initial scan of the tissue. Thecontroller (not shown) obtains actual values of the VOI and determinesif the actual values of VOI are within an acceptable initial range. Ifnot, then the controller (not shown) iteratively adjusts circuit valuesand performs post-simulation and re-computation until the actual VOI arewithin the acceptable range. This iterative process constitutes a secondoptimization process. If, however, the controller (not shown) determinesthat the actual VOI values are acceptable, then it begins the scanningof the tissue in a continuous manner, immediately or after receivinginput from a user. During the scanning of the tissue, the controller(not shown) continuously obtains actual VOI values and performs a thirdoptimization by adjusting circuit values to maintain the actual VOIvalues within an acceptable operational range. This third optimizationmay be necessary if the subject moves or there is parametrical drift.

Referring to FIG. 9B, the steps associated with simulation and theinitial optimization are shown in more detail. The block diagram of FIG.9B begins with the step of “Build 3D CAD model with same dimension andmaterial as physical coils.” This step is directed to coil computeraided design (CAD) models which refer to the 3D models of the coil, withthe same material properties and dimension as the physical coil. Thecoil parameters include coil capacitor, inductor values and othercircuit parameters known to a person having ordinary skill in the art.These parameters and models are thus loaded into a simulator (commercialsimulators are available, as well as application specific simulators) inorder to carry out a full-wave electromagnetic simulation. Next, atissue model of a tissue of interest (human or animal), e.g., ELLA,LOUIS, HANAKO, or DUKE, is loaded into the simulator. Next, the discreteports are setup by connecting the edges in the coil CAD model where thecapacitor, inductor or power input are connected with voltage/currentsource, and where each source is associated with a corresponding port inthe simulation. Next, the full-wave simulation is carried out to obtainmagnetic field, associated electric field, and S-parameters for eachport. Next, the S-parameter for each port is examined to determine ifthe S-parameter is within an acceptable range of benchtop measurementsof the actual coil. If not, then the method of the present disclosureiteratively returns to the step of setting up discrete port and rerunsthe simulation. Once the S-parameters are within an acceptable range ofbenchtop measurements, then the method exports S-parameters, B and theassociated E fields to an optimization engine. The optimization engineloads the S-parameters of each coil (simulated or actual), and assignsport parameters from the full-wave electromagnetic simulation. Next, themethod defines the lumped circuit parameters for each port for theoptimization engine. Next, boundaries for the lumped circuit parametersare defined based on a predefined set of boundaries. Next, the methodloads B and associated E fields from the full-wave electromagneticsimulation software into optimization engine. Next, target S-parameteris provided to the optimization engine. Next, for the optimizationengine excitation voltage for each transmit coil is assigned. Next, theoptimization engine begins its optimization with optimization settingsincluding choice of optimization solvers, optimization steps size, howmany steps and initial values, as known to a person having ordinaryskill in the art. The optimization engine continues its optimizationuntil it converges to a solution which is when the cost function (seeEq. 11) reaches a stable value. It should be noted that in equation 11,the S-parameter (S) and B1 fields (B1) are all functions of x, and x isa vector of coil parameters such as capacitor values (C), inductorvalues (L) and decoupling circuit parameters such as couplingcoefficient (k) and quality factor (Q). In the optimization, the methodof the present disclosure is optimizing x with a choice of step size ofdelta x. After, e.g., 10000 steps, the cost function f(x) becomes astable value, at which point the optimization is considered as havingconverged. Furthermore, it should be noted that in equation 11, Sii, Sijare given by the users as the target S parameters. And the “target” inequation 11 is the B1 homogeneity (or B1 coefficient of variation) as atarget value given by the user (i.e., VOI). Once the cost function f(x)is stable, the diag(S(x)) should be close to S_(ii), S_(r)(x) should beclose to S_(ii) and Std(B1(x))/Mean(B1(x)) should be close to the targetB₁ ⁺homogeneity. Next, different VOIs (cross-talk, B₁ ⁺ homogeneity, B₁⁺ power efficiency, and any combination thereof) are tested to determineif within an acceptable range of an associated target. If no, then themethod moves back to the step of lumped circuit parameters andreadjusting those. If yes, then the method moves the established B andthe associated E field for all the ports, and proceeds to superpositionB and E fields of all ports to generate B and E fields for each coil.Next the method generates specific absorption rate (SAR) prediction forregion of interest (as previously identified ROI) as well as virtualobservation points using the E-fields.

Referring to FIG. 10 , an example of a computer system that constitutesthe controller (not shown) discussed with reference to FIG. 9A isprovided that can interface with the above-discussed MRI system.Referring to FIG. 6 , a high-level diagram showing the components of anexemplary data-processing system 1000 for analyzing data and performingother analyses described herein, and related components. The systemincludes a processor 1086, a peripheral system 1020, a user interfacesystem 1030, and a data storage system 1040. The peripheral system 1020,the user interface system 1030 and the data storage system 1040 arecommunicatively connected to the processor 1086. Processor 1086 can becommunicatively connected to network 1050 (shown in phantom), e.g., theInternet or a leased line, as discussed below. The imaging described inthe present disclosure may be obtained using imaging sensors 1021 and/ordisplayed using display units (included in user interface system 1030)which can each include one or more of systems 1086, 1020, 1030, 1040,and can each connect to one or more network(s) 1050. Processor 1086, andother processing devices described herein, can each include one or moremicroprocessors, microcontrollers, field-programmable gate arrays(FPGAs), application-specific integrated circuits (ASICs), programmablelogic devices (PLDs), programmable logic arrays (PLAs), programmablearray logic devices (PALs), or digital signal processors (DSPs).

Processor 1086 can implement processes of various aspects describedherein. Processor 1086 can be or include one or more device(s) forautomatically operating on data, e.g., a central processing unit (CPU),microcontroller (MCU), desktop computer, laptop computer, mainframecomputer, personal digital assistant, digital camera, cellular phone,smartphone, or any other device for processing data, managing data, orhandling data, whether implemented with electrical, magnetic, optical,biological components, or otherwise. Processor 1086 can includeHarvard-architecture components, modified-Harvard-architecturecomponents, or Von-Neumann-architecture components.

The phrase “communicatively connected” includes any type of connection,wired or wireless, for communicating data between devices or processors.These devices or processors can be located in physical proximity or not.For example, subsystems such as peripheral system 1020, user interfacesystem 1030, and data storage system 1040 are shown separately from thedata processing system 1086 but can be stored completely or partiallywithin the data processing system 1086.

The peripheral system 1020 can include one or more devices configured toprovide digital content records to the processor 1086. For example, theperipheral system 1020 can include digital still cameras, digital videocameras, cellular phones, or other data processors. The processor 1086,upon receipt of digital content records from a device in the peripheralsystem 1020, can store such digital content records in the data storagesystem 1040.

The user interface system 1030 can include a mouse, a keyboard, anothercomputer (connected, e.g., via a network or a null-modem cable), or anydevice or combination of devices from which data is input to theprocessor 1086. The user interface system 1030 also can include adisplay device, a processor-accessible memory, or any device orcombination of devices to which data is output by the processor 1086.The user interface system 1030 and the data storage system 1040 canshare a processor-accessible memory.

In various aspects, processor 1086 includes or is connected tocommunication interface 1015 that is coupled via network link 1016(shown in phantom) to network 1050. For example, communication interface1015 can include an integrated services digital network (ISDN) terminaladapter or a modem to communicate data via a telephone line; a networkinterface to communicate data via a local-area network (LAN), e.g., anEthernet LAN, or wide-area network (WAN); or a radio to communicate datavia a wireless link, e.g., WiFi or GSM. Communication interface 1015sends and receives electrical, electromagnetic or optical signals thatcarry digital or analog data streams representing various types ofinformation across network link 1016 to network 1050. Network link 1016can be connected to network 1050 via a switch, gateway, hub, router, orother networking device.

Processor 1086 can send messages and receive data, including programcode, through network 1050, network link 1016 and communicationinterface 1015. For example, a server can store requested code for anapplication program (e.g., a JAVA applet) on a tangible non-volatilecomputer-readable storage medium to which it is connected. The servercan retrieve the code from the medium and transmit it through network1050 to communication interface 1015. The received code can be executedby processor 1086 as it is received, or stored in data storage system1040 for later execution.

Data storage system 1040 can include or be communicatively connectedwith one or more processor-accessible memories configured to storeinformation. The memories can be, e.g., within a chassis or as parts ofa distributed system. The phrase “processor-accessible memory” isintended to include any data storage device to or from which processor1086 can transfer data (using appropriate components of peripheralsystem 1020), whether volatile or nonvolatile; removable or fixed;electronic, magnetic, optical, chemical, mechanical, or otherwise.Exemplary processor-accessible memories include but are not limited to:registers, floppy disks, hard disks, tapes, bar codes, Compact Discs,DVDs, read-only memories (ROM), erasable programmable read-only memories(EPROM, EEPROM, or Flash), and random-access memories (RAMs). One of theprocessor-accessible memories in the data storage system 1040 can be atangible non-transitory computer-readable storage medium, i.e., anon-transitory device or article of manufacture that participates instoring instructions that can be provided to processor 1086 forexecution.

In an example, data storage system 1040 includes code memory 1041, e.g.,a RAM, and disk 1043, e.g., a tangible computer-readable rotationalstorage device such as a hard drive. Computer program instructions areread into code memory 1041 from disk 1043. Processor 1086 then executesone or more sequences of the computer program instructions loaded intocode memory 1041, as a result performing process steps described herein.In this way, processor 1086 carries out a computer implemented process.For example, steps of methods described herein, blocks of the flowchartillustrations or block diagrams herein, and combinations of those, canbe implemented by computer program instructions. Code memory 1041 canalso store data, or can store only code.

Various aspects described herein may be embodied as systems or methods.Accordingly, various aspects herein may take the form of an entirelyhardware aspect, an entirely software aspect (including firmware,resident software, micro-code, etc.), or an aspect combining softwareand hardware aspects. These aspects can all generally be referred toherein as a “service,” “circuit,” “circuitry,” “module,” or “system.”

Furthermore, various aspects herein may be embodied as computer programproducts including computer readable program code stored on a tangiblenon-transitory computer readable medium. Such a medium can bemanufactured as is conventional for such articles, e.g., by pressing aCD-ROM. The program code includes computer program instructions that canbe loaded into processor 1086 (and possibly also other processors), tocause functions, acts, or operational steps of various aspects herein tobe performed by the processor 1086 (or other processors). Computerprogram code for carrying out operations for various aspects describedherein may be written in any combination of one or more programminglanguage(s), and can be loaded from disk 1043 into code memory 1041 forexecution. The program code may execute, e.g., entirely on processor1086, partly on processor 1086 and partly on a remote computer connectedto network 1050, or entirely on the remote computer.

It should be noted that while the present disclosure make references totransceivers, implying a coil array that performs both transmit andreceive, coil arrays that are configured to only perform transmit, aswell as transmit arrays are within the ambit of the present disclosure.

Those having ordinary skill in the art will recognize that numerousmodifications can be made to the specific implementations describedabove. The implementations should not be limited to the particularlimitations described. Other implementations may be possible.

1. A method of operating a multi-coil magnetic resonance imaging system,comprising: a controller having a processor and software loaded ontangible memory performing a simulation using a predefined tissue modelfor a tissue to be imaged; the controller determining output values of avariable of interest (VOI) associated with operation of two or morecoils of a magnetic resonance imaging system based on the simulation;the controller comparing the simulated output values of the VOI to an apriori target values of the VOI; if the simulated output values of theVOI are outside of a predetermined envelope about the a priori targetvalues of the VOI, then the controller performing a first optimization,wherein the first optimization includes: iteratively adjusting thecircuit values until the simulated output values of the VOI are withinthe predetermined envelope about the a priori target values of the VOIthereby establishing VOI optimized values; and loading the establishedVOI optimized values and operating the magnetic resonance imaging systemon the tissue to be imaged.
 2. The method of claim 1, wherein the firstoptimization is based on a cost function defined as:${f(x)} = {{w_{1}{{{❘{{diag}\left( {S(x)} \right)}❘} - S_{ii}}}} + {w_{2}{{{❘{S_{r}(x)}❘} - S_{ij}}}} + {w_{3}{{\frac{S{{td}\left( B_{1}^{+} \right)}}{{Mean}\left( B_{1}^{+} \right)} - {target}}}}}$wherein ƒ(x) is the cost function of a real number vector x whoseentries are coil parameters, ∥ ∥ denotes the Euclidean distance, | | isthe elementwise absolute values, w₁₋₃ are weights, B₁ ⁺ represents fieldassociated with each coil of the two or more coils, and targetrepresents target values for the variable of interest.
 3. The method ofclaim 2, wherein the VOI is cross-talk between any two neighboringcoils.
 4. The method of claim 2, wherein VOI is B₁ ⁺ homogeneity.
 5. Themethod of claim 2, wherein VOI is B₁ ⁺ power efficiency
 6. The method ofclaim 2, wherein VOI is a combination of cross-talk between any twoneighboring coils and B₁ ⁺homogeneity.
 7. The method of claim 2, furthercomprising: exporting one or more of the circuit values, magnetic andelectric field values as well as S-parameters of each of the two or morecoils; performing an initial scan of a tissue of interest associatedwith the tissue model; obtaining actual output values of the VOI; if theactual output values of the VOI of the initial scan are outside of apredetermined envelope about the a priori target values of the VOI, thenperforming a second optimization, wherein the second optimizationincludes: establishing an error parameter based on the actual measuredVOI from the initial scan vs. the loaded target values, and iterativelyadjusting the circuit values using a first gradient descent optimizationprocess, re-computing the VOI, and re-comparing the recomputed values ofthe VOI to the loaded target values of the VOI until the recomputed VOIvalues are within the predetermined envelope about the a priori targetvalues of the VOI.
 8. The method of claim 7, wherein the VOI is one ofcross-talk, B₁ ⁺homogeneity, B₁ ⁺ power efficiency, and any combinationthereof.
 9. The method of claim 7, further comprising: exporting one ormore of the post circuit values, magnetic and electric field values aswell as S-parameter of each of the two or more coils; continuouslyoperating the two or more coil; obtaining continuous actual outputvalues of the VOI; if the continuous actual output values of the VOI areoutside of a predetermined operational envelope about the a prioritarget values of the VOI, then performing a third optimization, whereinthe third optimization includes: iteratively adjusting the circuitvalues using a second gradient descent optimization process,continuously operating the two or more coils, and re-comparing theactual output values of the VOI to the a priori target values of the VOIuntil the actual output values are within the predetermined operationalenvelope.
 10. The method of claim 9, wherein the VOI is one ofcross-talk, B₁ ⁺homogeneity, B₁ ⁺ power efficiency, and any combinationthereof.
 11. A drive system for a multi-coil magnetic resonance imagingsystem, comprising: two or more coils utilized for imaging a tissue ofinterest; a drive circuit for driving the two or more coils; acontroller having a processor and software loaded on tangible memoryadapted to: perform a simulation using a predefined tissue model for atissue to be imaged determine output values of a variable of interest(VOI) associated with operation of two or more coils of a magneticresonance imaging system based on the simulation; compare the simulatedoutput values of the VOI to an a priori target values of the VOI; if thesimulated output values of the VOI are outside of a predeterminedenvelope about the a priori target values of the VOI, then perform afirst optimization, wherein the first optimization includes: iterativelyadjust the circuit values until the simulated output values of the VOIare within the predetermined envelope about the a priori target valuesof the VOI thereby establishing VOI optimized values; and load theestablished VOI optimized values and operate the magnetic resonanceimaging system on the tissue to be imaged.
 12. The system of claim 11,wherein the first optimization is based on a cost function defined as:${f(x)} = {{w_{1}{{{❘{{diag}\left( {S(x)} \right)}❘} - S_{ii}}}} + {w_{2}{{{❘{S_{r}(x)}❘} - S_{ij}}}} + {w_{3}{{\frac{S{{td}\left( B_{1}^{+} \right)}}{{Mean}\left( B_{1}^{+} \right)} - {target}}}}}$wherein ƒ(x) is the cost function of a real number vector x whoseentries are coil parameters, ∥ ∥ denotes the Euclidean distance, | | isthe elementwise absolute values, w₁₋₃ are weights, B₁ ⁺ represents fieldassociated with each coil of the two or more coils, and targetrepresents target values for the variable of interest.
 13. The system ofclaim 12, wherein the VOI is cross-talk between any two neighboringcoils.
 14. The system of claim 12, wherein VOI is B₁ ⁺ homogeneity. 15.The system of claim 12, wherein VOI is B₁ ⁺ power efficiency
 16. Thesystem of claim 12, wherein VOI is a combination of cross-talk betweenany two neighboring coils and B₁ ⁺homogeneity.
 17. The system of claim12, the controller further adapted to: export one or more of the circuitvalues, magnetic and electric field values as well as S-parameter ofeach of the two or more coils; perform an initial scan of a tissue ofinterest associated with the tissue model; obtain actual output valuesof the VOI; if the actual output values of the VOI of the initial scanare outside of a predetermined envelope about the a priori target valuesof the VOI, then performing a second optimization, wherein the secondoptimization includes: establish an error parameter based on the actualmeasured VOI from the initial scan vs. the a priori target values, anditeratively adjust the circuit values using a first gradient descentoptimization process, re-compute the VOI, and re-compare the recomputedvalues of the VOI to the a priori target values of the VOI until therecomputed VOI values are within the predetermined envelope about the apriori target values of the VOI.
 18. The system of claim 17, wherein theVOI is one of cross-talk, B₁ ⁺homogeneity, B₁ ⁺ power efficiency, andany combination thereof.
 19. The system of claim 17, the controllerfurther adapted to: export one or more of the post circuit values,magnetic and electric field values as well as S-parameter of each of thetwo or more coils; continuously operate the two or more coil; obtaincontinuous actual output values of the VOI; if the continuous actualoutput values of the VOI are outside of a predetermined operationalenvelope about the a priori target values of the VOI, then performing athird optimization, wherein the third optimization includes: iterativelyadjust the circuit values using a second gradient descent optimizationprocess, continuously operate the two or more coils, and re-compare theactual output values of the VOI to the a priori target values of the VOIuntil the actual output values are within the predetermined operationalenvelope.
 20. The system of claim 19, wherein the VOI is one ofcross-talk, B₁ ⁺ homogeneity, B₁ ⁺ power efficiency, and any combinationthereof.